One Shot One Kill

 Author: Kevin Tucker Unit Title: One Shot One Kill Grade: 12 Subject: Pre-Calculus Estimated Duration: 7 - 48 Minute Class Period Unit Activities: Background Knowledge:  Students studied the quadratic equation and the quadratic formula in Algebra I and Algebra II.  The students have also played games that involve throwing an object.   Date: July 2013

The Big Idea (including global relevance)

To use parametric equations and component vectors to calculate the angle needed to launch a projectile and to hit a target at a known distance.  In military applications we need to hit our desired target the first time.

The Essential Question

1. Can we launch a projectile and hit a target at a known distance?

Justification for Selection of Content

Modeling projectile motion involves parametric equations that breaks velocity into its horizontal and vertical components.  In order to calculate the angle needed to hit a target at a known distance the students will have to solve systems of equations with a linear equation and a quadratic equation.  They will also have to use inverse trig functions to solve for the angle.

The Challenge

To create a way for a team of 3 or 4 students to use a 3-man water balloon slingshot to hit a target at a known distance the first time.  Groups will need to come up with a way to keep initial height, angle and velocity constant.

The Hook

Videos of projectile motion in war.  Have Mr. Duke former USMC sniper (current PHS teacher talk to students) if schedule permits.

Teacher's Guiding Questions

1.     What do we need to know in order to hit a target?

2.     When testing our launcher and collecting data what things do we need                to keep constant?

3.     What are some factors that are out of your control and how can you                       minimize them?

4.     How can we model the motion of our projectile?

5.     What do we need to do to solve for the angle?

ACS (Real world applications; career connections; societal impact)

A – (Real World Application):  Hitting a target at a known distance is applicable in hunting, military and for fun when launching water balloons in the summer.  Projectile motion can also be applied to any sport with a ball (golf, basketball, baseball etc.).

C – (Career Connections): Military, Police, Sports.

S – (Societal Impact):  In a Military situation miscalculations would result in the death of innocent civilians or fellow soldiers (friendly fire).

Engineering Design Process (EDP)

Students will have to design a process to calculate the velocity of the water balloon.  They will have to think about how to keep the velocity as constant as possible.

Students will have to use the engineering design process to develop a way for the team of 3 or 4 students to hit a target at a known distance using a 3-man slingshot.  In order to hit the target the students will have to design a way to hold the ends of the sling shot.  Pull back the slingshot.  Keep the velocity constant (same stretch), adjust the angle, and aim.

Where the CBL and EDP appear in the Unit

On the first day the students will be presented with the challenge.  This will start the engineering design process.  There are essential 2 challenges. The first is to come up with a process to calculate the initial velocity.  The second is to come up with a way for the students to hit the target at a known distance.  This will require the students to come up with a process or structure to keep the velocity constant, keep the initial height constant, and calculate the angle.

Challenge 1 (calculate the initial velocity) Activity 4.

Identify Problem:

The students will identify the problem.  What is the initial velocity of the projectile?

Brainstorm ideas:

At this stage the students will have to think about how to calculate velocity (distance/ time).  In this there are multiple options.  The easiest is to launch the water balloons horizontally.  Thus velocity is distance over time.  If the launch it at an angle they will have to take the angle into account.  If they fix the initial height they can solve for the time to hit the ground and eliminate the need for a stop watch.

Design:

The groups will design their process, then go to the field and calculate their initial velocity.  This answer will be used for the next challenge.  The students may need to revise their design if they do not get a reasonable value for the initial velocity.

Challenge 2 (hitting the target at known distances) Activity 6.

Identify the problem:

To hit a target at a known distance.  Students will need to think about how their group of 3 to 4 will manage controlling the initial height, the initial velocity, and the angle.

Brainstorm Ideas:

The students will brainstorm ideas.  How do they want to fix the height (build a structure, have 2 people hold the ends.  How will they keep the velocity constant have 2 students measure the distance of the stretch, or create a way to make sure it stretches the same amount every time.  How will they measure the angle?

Design:

Students will design their process or structure and build it.  We will then go to the field and the students will implement their design and try to hit the targets at 90 ft., 180 ft., and 270 ft.

Activity 7

Share solution: The groups will present their solutions to the class.

Misconceptions

This will be the first time students have used horizontal and vertical components to model projectile motion.  In Algebra I and II students only looked at problems where the ball was thrown straight up. This limits their ability to solve realistic problems.

HSN-VM.A.2 – Represent and model with vector quantities – find the components of a vector.

HSN-VM.A.3 – Represent and model with vector quantities – Solve problems involving velocity and other quantities that can be represented by vectors.

HSF-IF.C7 – Analyze functions using different representations.

HSF-TF.B.7 – Model periodic phenomena with trigonometric functions.

HSF-TH.C.9 – Prove and apply trigonometric identities.

Assesments
Pre & Post-Test Results

After analyzing the Pre-Assessment and the Post-Assessment data I noticed that the grades improved dramatically.  The Assessment was 6 questions 5 5 point questions and a 1 point extension question.  The average score on the Pre-Assessment was a 20% and the average score on the post assessment was a 70%.  This was a 250% increase.  Unfortunately the seniors in the class graduated before we could finish the project so they did not take the Post-Assessment.  The question that students had the most issue with was finding how high the balloon goes.  We did not spend much time in class talking about finding the vertex of the y component of the parametric equation due to the fact that we ran out of time at the end of the school year.
How to Make This a Hierarchical Unit

In a middle school class you could have students play with the angle and velocity of the object and allow them to make predictions.  This would be a great introduction into non-linear functions.

http://phet.colorado.edu/en/simulation/projectile-motion this website is a great simulation program.

Reflection

The success of the lesson is that the students had the opportunity to apply mathematical concepts and see how the math works in the real world.  Since this was the third unit I taught from the CEEMS project I felt that going over the essential and guiding questions went great this time around.  I had the students work in their groups to come up with a list of questions.  I gave the students about 8 minutes to come up with their list of questions and had each group share 2 or 3 of their questions.  As a class we decided if it is an essential or a guiding question.  After each group shared I asked if there were any other questions they would like to add.  This worked better because each group was forced to contribute something to the conversation and by allowing the students to add at the end everyone could have their ideas heard.

Another success of the lesson was that the groups that built a structure to fix the initial height were much more successful at getting close to the target.  One group missed the target by 3 feet at 45 yards, by 10 feet at 60 yards and by 5 feet at 90 yards.  The students also had a lot of fun launching water balloons at their teacher at the end of the school year.

A shortcoming of this lesson was that we physically ran out of time at the end of the school year.  The seniors graduated before they could take the post assessment.  As I have found will all of my CEEMS lessons time is always a factor.

After talking with the juniors after the project was over they suggested that I require that all groups make some sort of a structure.  Some groups tried to hold the ends of the slingshots and found that it was difficult to try to control the stretch, the height and the angle while holding it under tension.  They also saw how the groups that built structures were more accurate.  I also had a group of all girls that did not have the strength to hold the slingshot and pull it back (they said NO ALL GIRL GROUPS for the project.)